In class, we explored the representation of marriage through bipartite graphs. After going through the process with the males proposing, we repeated with the females proposing. In both cases, we found matches that satisfied all the players of the game. Of course, when males propose, they are better off, and when females propose, they are better off. However, the two iterations yielded two different satisfactory outcomes, which destroys the notion that there is only one match for each person. Despite the fact that in reality, the love game is not solely based on preferences as it was in class, there is much general debate over the idea of "the one." If our world was simplified down to mathematical bipartite graphs, so many distressed seekers of their "soul mate" would be relieved.

so i was watching "how i met your mother" the other day and it was the episode where ted (and barney) go to a matchmaking service that uses a computer to match people up based on compatibility, traits, etc, claiming a 100% success rate. it then generates a compatibility score 1-10 (10 highest).

anyway the computer finds only one good result for ted, but she is already engaged. this girl is an 8.5 match with her fiancee but her compatibility score with ted is 9.6. ted runs off to find her, exclaiming "but 9.6 is 11.45% better than 8.5! would you turn down an 11.45% improvement?" his efforts fail, as she becomes happily married.

this episode debunks the whole idea of matching through numerical preferences, at least in love.

As in the TV episode you mentioned, the bipartite graphs we looked at in class failed to account for the fact that not all people will take an improvement over a sure thing.

For example, in the bipartite graph we looked at in class, the women would get asked by the men. If they were later asked by a man higher on their list, they would dump the first one for the higher-ranked man. However, this is not as realistic as we were lead to believe in class. A lot of people would rather not break an engagement in favor of a "better" man or woman because they are too nice to let that person down. Or, they could associate a price with the hassle of breaking an engagement, canceling wedding plans, and restarting with a new person. If these costs outweigh the benefit gained from the new partner, they would rather stick with the first, less compatible person.

This disproves the method we used in class as an approximation even though that method is generally true (that a lot of people would give up a partner for a better one regardless of any other factor).

It also made me think about feminism's role as it relates to the divorce rate. In the 1930s when the divorce rate was lower than it is now, women were generally less independent than they are today. Maybe now that men and women are asking each other to marry, more men are not getting their top choices and are less satisfied with their wives than their fathers were. I don't mean to be sexist in any way, I just thought it was an interesting observation. If there is a dominant gender and a gender that doesn't mind getting screwed, then the "one gender asking another" model is more accurate. However, in a society like we have today, there are some women who do "propose" first and thus the model we used in class would have to be changed to reflect this.

What about the possibility of some matches in a bipartite graph staying the same regardless of which direction you work from, either men initiating or women initiating? Would that not be "the one" in some way? Even though they might not be the top of the other's list, if they're stable regardless of who initiates that should speak for something. And this is just with sticking within the model.

But as others have pointed out, the bipartite model we've studied in class is fail. It's too static, just a one-and-done preference list and the resulting pairings. These preference lists change over time through experience, and if some model were to take that into account, I would argue that these preference lists would converge at least somewhat to a point where choices are mutual and not dependent on who goes first.

I think that the preference list is valid. From what I've seen in people, that's not an inaccurate way of describing preferences. And remember, there was a tentative engagement, not the sort we have nowadays. But if you look at it in terms of dating and relationships, people run off with other people, breaking off an existing relationship sometimes. The difference is that it is random when a node will "propose" to another node to start dating. If neither node asks the other out, waiting for the other one to, then when someone else asks one of them out, then they end up settling for less, as does happen. But as far as soul mates, I don't know if that's how it works. I think that there are people who are compatible with each other on deep levels, but with billions of people in the world, it is unlikely that there is only one single person designed for someone else. Also, there is not a perfect gender balance in the world. In China, there is a very high male female ratio, for instance.